Huff Garza
03/10/2024 · Middle School
Use the change-of-base formula and a calculator to evaluate the logarithm. \( \log _{7} 42 \) \( \log _{7} 42 \approx \square \) (Do not round until the final answer. Then round to the nearest thousandth as needed.)
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Tutor-Verified Answer
Step-by-step Solution
To evaluate \( \log_{7} 42 \) using the change-of-base formula, we can use the following formula:
\[
\log_{b} a = \frac{\log_{c} a}{\log_{c} b}
\]
where \( c \) can be any positive number (commonly 10 or \( e \)). Here, we will use base 10.
Thus, we have:
\[
\log_{7} 42 = \frac{\log_{10} 42}{\log_{10} 7}
\]
Now, we can calculate \( \log_{10} 42 \) and \( \log_{10} 7 \) using a calculator:
1. Calculate \( \log_{10} 42 \):
\[
\log_{10} 42 \approx 1.6232
\]
2. Calculate \( \log_{10} 7 \):
\[
\log_{10} 7 \approx 0.8451
\]
Now, substitute these values into the change-of-base formula:
\[
\log_{7} 42 \approx \frac{1.6232}{0.8451}
\]
Now, perform the division:
\[
\log_{7} 42 \approx 1.920
\]
Finally, rounding to the nearest thousandth, we have:
\[
\log_{7} 42 \approx 1.920
\]
Thus, the final answer is:
\[
\log_{7} 42 \approx 1.920
\]
Quick Answer
\[
\log_{7} 42 \approx 1.920
\]
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